-
Fundamental J. Modern Physics, Vol. 1, Issue 2, 2011, Pages
223-246 Published online at http://www.frdint.com/
:esphras and Keywords negative indexed material (NRM),
meta-material, group refractive
index, phase refractive index, phase momentum, wave momentum,
group and phase velocity, mechanical momentum, reactive energy.
*Corresponding author
Received May 10, 2011; Revised July 20, 2011
© 2011 Fundamental Research and Development International
A NEW MECHANICS OF CORPUSCULAR-WAVE TRANSPORT OF MOMENTUM AND
ENERGY INSIDE
NEGATIVE INDEXED MATERIAL
SHANTANU DAS1,*, SOUGATA CHATERJEE2, AMITESH KUMAR2, PAULAMI
SARKAR2, ARIJIT MAZUMDER2, ANANTA LAL DAS2 and SUBAL KAR3
1Reactor Control Division Bhabha Atomic Research Centre (BARC)
Mumbai, India e-mail: [email protected]
2Society for Applied Microwave Electronics Engineering and
Research (SAMEER) Kolkata, India
3Institute of Radio Physics and Electronics (IRPE) University of
Calcutta Kolkata, India
Abstract
A century has passed regarding wave-particle duality, well an
electromagnetic (EM) radiation in dispersionless free space vacuum
is represented by a photon, with corpuscular and wave nature. The
discussions from the past century aimed at the nature of photon
inside a media having dispersion in the refraction property, other
than free space.
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SHANTANU DAS et al.
224
We call mechanical momentum, wave-momentum, and try to match our
‘thought experiments’ with intriguing property of this ‘photon’ or
pulse carrying EM energy packet, and more so we try to find its
property energy, momentum inside a media, a positive refractive
media. Well if the media show a negative refractive index behavior,
then these queries are profound, and suitable explanations to these
classical concepts of corpuscular-wave nature of photon inside
these media are quest for the scientists dealing with these
meta-materials. Here some of this counterintuitive nature of
corpuscular-wave nature of photon inside negative indexed material
is brought out, with possible ‘new definition’ of its
‘wave-momentum’, the concept of ‘reactive energy’ inside negative
indexed material, along with possible ‘new wave equation’. These
definitions and expressions of ‘wave-momentum’ and ‘reactive
energy’ pertaining to negative indexed material are new and
discussed and derived by classical means.
1. Introduction
We have demonstrated negative refractive index ‘meta-material’
plasmonic structures in Ka-band. In our experimental investigation,
we have made these plasmonic meta-material prisms of 45, 30 and 15
degrees to get enhanced transmittance of more than 15 dB from
background; at negative angles indicating a refractive index of
about –1.8, [20-26]. This paper is not aimed for this experimental
design, where the meta-material realized by us is based on simple
wire-array and labyrinth resonators, [20-26], but to focus on
possible theory of the wave mechanics coupled to particle nature of
the EM radiation, energy and momentum transport anomalies, a
possible new momentum energy description. Also in our repeated
observations on numerical experiments we get, as to if a pulse of
EM radiation is launched inside a negative refractive index
material (NRM), gets squeezed sharpened, [20-26] (refer Figure 1)
similarly a spherical wave front in positive indexed media gets
flattened as it propagates inside the NRM [20-26]. Though several
approaches to explain these counterintuitive phenomena have been
evolved, yet it is interesting if in the meta-material parlance
particle-wave theory be founded! Here we give possible classical
explanations to these counterintuitive phenomena and also new
explanations regarding energy momentum, wave equation if applied to
this negative indexed material: how shall they look, vis-à-vis
positive indexed systems. This problem is a topic subject of
investigation in modern optics also. We
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propose the concept of reactive energy and expression for new
wave-momentum for pulse of electromagnetic energy inside a medium
(negative refractive indexed), with suitable derivation along with
new wave equation. The research papers [7-10, 12, 27] discussed
momentum and energy of this reversed electrodynamics in other
contexts. Herein, we are deriving the similar concepts with
different approach, limiting to propagation of EM pulse inside
NRM.
2. Phase and Group Refractive Index in Negative Refractive Index
Material
Let us demarcate the two refractive indices, [1, 2], and [3, 11,
19] and this demarcation is essential in explaining the NRM theory.
Take the refractive index dispersive that is a function of
frequency call it ( ) ,ωpn phase refractive index. This
is basic refractive index by which the velocity of phases of
travelling gets modulated inside a dispersive media. We call it
phase index .pn Similarly velocity of a group
of frequency travelling wave gets modulated in the media that
gives group refractive index .gn In case of NRM, the phase
refractive index if it were ( ) 10 −=ωpn at a
particular frequency ,0ω it would imply that in that media the
phases would be
travelling with speed of light but in opposite direction. There
is a backward wave inside NRM [1-3, 11, 13-18, 25, 26]. Refer
Figure 1C; where it is demonstrated that phase gets reversed while
inside NRM compared to the free space propagation. Now if there is
no change in the refractive index for phases with respect to
frequency, meaning that { ( ) } ,0=ωω ddn p we call it
dispersionless medium. In that case the
phase velocity ( )ωpv of the wave and group velocity ( )ωgv of
the wave are same.
In the free space (refer Figure 1A) both group of frequencies
and the crests and troughs of phases are travelling with ( ) ( )
.00 cvv gp =ω=ω In the free space, we
have same modulation for the phases of the signal and group of
frequencies at a particular frequency and thus we say phase and
group index are same
( ) ( ) .100 =ω=ω gp nn If the media were dispersive, we take
phase refractive
index as an ‘analytic’ function of the frequency, that is, ( )ω=
analyticfn p at a
particular frequency .0ω Expansion of Taylor [1-3, 11, 13-18,
25, 26], series (1) for
this dispersive phase refractive index; taking the origin at
,0ω=ω that is frequency
of NRM behavior, (only to its first derivative term at the
frequency 0ω near electric
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SHANTANU DAS et al.
226
plasma and magnetic plasma resonance, where 0ωgv are
always positive [1-3, 11, 13-18, 25, 26].
In the introduction, we have made a statement of our prism
experiment showing a negative value of refractive index of –1.8. We
clarify that the observed negative refraction is for
‘phase-refractive-index’ as ( ) ,8.10 −≅ωpn at ,GHz3320 ≅πω
with region of NRM as GHz,85.0≅ω∆ whereas the group refractive
index
( ) ,00 >ωgn as this gives positive group velocity. We thus
can say that we can
observe a negative phase refractive index but the group
refractive index shall always be positive. Equation (1) should be
read at a particular frequency 0ω of interest,
where we are observing a negative refractive index, in our
experimental case it were around 33 GHz, [25].
We can emulate and model by a simplest model as in (2). An NRM
(phase refractive index), by a function such that 0ω is a frequency
below which the phase
refractive index is negative and above which the phase
refractive index is positive. [1-3, 11, 13-18, 25, 26], as (2)
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( ) .1 220
ω
ω−=ωpn (2)
This (2) is simplest form of model where one gets ENG (Epsilon
Negative) and
MNG (Mu Negative) material representation as ( ) ( )221 ωω−=ωε
epr and
( ) ( ),1 22 ωω−=ωµ mpr where epω and mpω are, respectively,
electric and
magnetic frequencies below which the permittivity and
permeability are, respectively, negative. In (2), 0ω is chosen in
the region where ( )0ωεr and
( )0ωµr both are negative so that ( ) .00
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SHANTANU DAS et al.
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‘packet’ of energy, in free space travelling with speed of light
,c (refer Figure 1A)
when entering the NRM with ,1−=pn will retard the wave-packets
speed to
,gnc in this case ,3c (refer Figures 1B and 1C) though the
direction of travel of
wave-packet, energy will be in same direction as was in free
space; but the phases crests and troughs will here start travelling
in opposite to free-space with velocity
.c− This is implication of the phase and group refractive index
in NRM. The implication at NRM boundary of these opposite phases
meeting will form a ‘cusp’ which will be oscillating at the
junction of NRM to the free space (refer Figure 1B) [1-3, 4-6, 11,
25, 26]. This phenomenon of retardation of the wave-packet envelope
and change of direction of travel of crest and trough the phase,
inside NRM gives the ‘pulse-sharpening’ effect, and flattening of
wave-front effect, what we have been observing in our experiments
[25], also [4, 5, 6] (refer Figure 1C).
The cusps at the NRM boundary are due to counter propagation of
the ‘phases’ of the waves inside and outside the NRM, they are
surface charges, and at the boundary electric field at this cusp
oscillates, [4-6, 13-18, 25, 26]; as two sets of impinging wave
fronts meet at the interface with ENG (Epsilon Negative
Material
).0
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229
{ } ( ) .4
exp 22
0 00 ⎥⎦
⎤⎢⎣
⎡−σ−πσ= ω−ω+ czteeEE ticiz (4)
The field incident at 0=z is adequately represented by complex
electric field as
( )[ ] ( )[ ]∫ ω−σω−ω−ω= tkzidEE expexp 2200in
[ ( )] ( ) .4
expexp 22
00 ⎥⎦
⎤⎢⎣
⎡−σ−−ω−σπ= cztcztiE (5)
The expression (4) is for travelling electric field that has two
parts. The phase part given inside the {} brackets, and multiplied
by Gaussian travelling envelope in
free space as [ ( ) ],4exp 22 czt −σ− having variance ,2σ i.e.,
the width of the packet (Full Width Half Maxima FWHM). The packet
is travelling from left to right thus phases (crest and trough are
translating in z+ -direction) with a phase velocity
,cv p = and the group, i.e., the envelope carrying the
information/energy is
travelling with group velocity cvg = in the same direction of z+
in free space
having ,1+== gp nn [19]. Refer Figure 1A, (4) is depicted there
travelling
towards right with envelope as dashed and phases as solid
lines.
We investigate what happens when this (4), (5) incident Gaussian
electromagnetic pulse enters a medium. This Gaussian pulse is
centered at angular frequency 0ω and we assume that this energy
beam is weakly focused so we take
spatial spread in only one dimension. The reflection and
refraction of electromagnetic waves at an interface are described
by Fresnel law. For normal incident [19], we have reflection
coefficient ( )ωρ and transmission coefficient ( )ωτ described as
(6),
[19]; both being function of frequency since impedance of media
is dispersive.
( ) ( ) ,2,00
0ZZ
ZZZZZ
+=ωτ
+−
=ωρ (6)
where εµ=Z is impedance of medium and 0Z is free space
impedance. Note
for a NRM with ,1−=ε=µ rr the ,0ZZ = the incident beam suffers
no reflection
and is 100% transmitted. The forms of reflected and transmitted
waves follow from
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SHANTANU DAS et al.
230
the spectrum of the incidence pulse (5) as (7) and (8),
[19].
[ ( ) ] ( ) ( )[ ],expexp 2200ref cztidEE +ωωρσω−ω−ω= ∫ (7)
[ ( ) ] ( ) ( { } )[ ].expexp 2200trans czntidEE p ω−ω−ωτσω−ω−ω=
∫ (8) It suffices for our purpose to assume that spectrum is narrow
so that we can
approximate ( )ωρ and ( )ωτ by their values at 0ω and ( )ωpn by
first two terms of
Taylor series expansion (1). This leads to simple Gaussian forms
for (7) and (8) as (9) and (10).
( ) [ ( )] ( ) ,4
expexp 22
000ref
⎥⎦
⎤⎢⎣
⎡+σ−+ω−σπωρ= cztcztiEE (9)
( ) [ ( )] ( ) .4
expexp 22
000trans
⎥⎦
⎤⎢⎣
⎡−σ−−ω−σπωτ= czntczntiEE gp (10)
For 100% transmission when 0ZZ = say for NRM when ,1−=µ=ε rr
with
( ) 10 −=ωpn and ( ) ,30 =ωgn we get 0ref =E since ( ) ( ) 1,0
00 =ωτ=ωρ and
transmitted field inside NRM is thus given below (11).
[ ( )] ( ) .34
expexp 22
00trans
⎥⎦
⎤⎢⎣
⎡−σ−+ω−σπ= cztczntiEE p (11)
5. Energy Momentum of Gaussian Electromagnetic Pulse
To this Gaussian pulse, there is a packet of energy ;0ω we can
associate
momentum c0ω with this pulse. The research papers [7-10, 12, 27]
discuss
momentum of energy of this reversed electrodynamics, in
different context but our approach and discussions are differently
oriented. Inside a medium, we can have scenario where the momentum
can have different interpretation if we say
cnp p 0ω= as phase ‘wave’ momentum inside medium, then if the
media has
,1−=pn we get confused by this negative momentum indicating a
decrease in
pressure for radiation of electromagnetic wave, when it strikes
a boundary. Well call
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231
this momentum cnp 0ω as ‘wave’ momentum, to distinguish from
‘mechanical’
momentum (12), (13) (containing group velocity and group index)
as, Minkowski [7], or Abraham [8];
,202
02
1 cnvcnnp pggpm ω=ω= (12)
.2002 cvcnp ggm ω=ω= (13)
These definitions of mechanical momentum ensure that they are
positive, inside NRM as well. Well these mechanical momentum
definitions (12) and (13) give us non-confusing thought that even
with 0
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SHANTANU DAS et al.
232
{ } ( ) ⎥⎦
⎤⎢⎣
⎡−σ−πσ= ω−ω− 2
20
photon 34
exp00 czteeEE ticizN (15)
is different from original (4), that is { }( )2
2
00 40phton cztticizP eeeEE
−σ−ω−ω+πσ=
in the free space. Equation (15) seems to suggest that the pulse
envelope and the phases travel are in opposite direction, this
packet need not be thus called a photon packet rather ‘negative’
photon packet! (refer Figure 1C).
Here we are visualizing that electromagnetic pulse (4) is a
‘photon’. Well this is how scientist describes a ‘packet of wave’
as unit energy ‘photon’. Equation (4) when we talk in limiting case
with 0→σ becomes singular and ideally representing ‘single-photon’
with frequency .0ω Well who has really seen a photon
a mathematical abstraction and can thus well be approximated as
in (4). Our argument of ‘negative’ photon stems from the fact that
had there be 100% reflection to (4), ( ) ,10 =ωρ then we get, a
packet of original photon as in (16), from (9) where the envelope
and phases are travelling in z− direction after hitting the
boundary at ,0=z thus retaining the character of original
photon.
( ) [ ( )] ( ) ⎥⎦
⎤⎢⎣
⎡+σ−+ω−σπωρ= 2
2000
ref4
expexp cztcztiEE
{ }( )
.2
2
00 40photon cztticizP eeeEE
+σ−ω−ω−πσ== (16)
Reflected photon is original photon as incident photon, while
transmitted photon inside NRM is ‘negative’ photon.
6. Photon Momentum Transfer to the Medium
Taking clue from the above discussion let us define phase
momentum, or wave-momentum of a photon packet as (17); this choice
will be clear as we proceed for proof below.
( )
,sgn 00def
cN
cnn
np
gp
pc
ω=
ω (17)
where .1sgn,0,1sgn,0 −= pppp nnnn
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Well if the photon is in free space, then (17) would be cpc 0ω=
or if it were
in our chosen NRM with 1−=pn and ,3=gn then inside NRM this
‘negative’
photon has wave-momentum as ( ) .31 0 cpc ω−= Well we could have
chosen
(17) to be as ( ) cnnp gpc 0ω= too, but the chosen square root
for gn will be
explained in the next section, by total energy balance
formulation.
We start our discussion of effect of our single photon entering
the medium from region of free space. If the photon is totally
reflected, then because of the momentum conservation it transfers
c02 ω momentum to the medium. If the photon passes
into the medium in that case momentum will be transferred to the
medium at the interface surface where there will be reflection and
transmission, the momentum transferred to surface is given as
( ) ,1 0media Tpc
Rp −ω
+= (18)
where the reflection probability R and transmission probability
T [19] with respect to free-space impedance 0Z and impedance of
medium Z are defined as
( )
.4
, 20
022
0
02
ZZ
ZZT
ZZZZ
R+
=τ=⎟⎠⎞
⎜⎝⎛
+−
=ρ= (19)
Putting (19) in (18) and using p of (17), we get the following
algebraic manipulations
( ) cN
ZZ
ZZcZZ
ZZc
p 020
002
0
00media 4 ω
+−
ω⎟⎠⎞
⎜⎝⎛
+−
+ω
=
( ) ccN
ZZZZ
cZZZZ
c00
20
002
0
00 42 ω−ω
+−
ω⎟⎠⎞
⎜⎝⎛
+−
+ω
=
( )( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
+
−−
++
ω−
ω= 2
0
20
20
000 412
ZZZZ
NZZZZ
cc
( )( )
( ) .12412 0020
000 TNccZZ
ZZN
cc+
ω−
ω=
++
ω−
ω= (20)
Therefore with the definition of wave-momentum as in (17), we
get momentum
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SHANTANU DAS et al.
234
transferred to the media, at the surface as
( )
.sgn
12 00media T
nn
ncc
pgp
p⎟⎟⎠
⎞⎜⎜⎝
⎛+
ω−
ω= (21)
Using the mechanical momentum definitions of (12) and (13), and
doing the same algebraic manipulations, we get the mechanical
momentums transferred to the medium at the surface as
,12 200media
1 Tcvn
ccp gpm ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+
ω−
ω=
.12 00media
2 Tcv
ccp gm ⎟⎟
⎠
⎞⎜⎜⎝
⎛+
ω−
ω= (22)
Well, all these momentums transferred to medium at the surface
of all types (21) and (22) reduce to c02 ω for a perfectly
reflecting surface when ,0=T
corresponding to change in momentum due to reflection. It is
also clear that mechanical momentum transferred to medium by
definition of 2mp will always be
positive as ,cvg < however the definition of ,1mp and cp
(17), when used the
momentum transfer to the medium at surface can be positive or
negative depending on the property of media.
Let us take an example of ideal case whence 0=R and ,1=T zero
reflection
and 100% transmission, for NRM with ,1−=pn ,3=gn and .3cvg =
The
condition for this is ,1−=µ=ε −− rr gives ,0ZZ = thus .0=R Here
the photon
passes into NRM with 100% probability ( ).1=T For this NRM
condition the
momentum transfer associated with mechanical momentums are
identical, corresponding to ( ) ,1 cvg− that is, 32 of the original
photon mechanical
momentum transferred to the media. The mechanical momentum
retained by photon is ( )31 the original photon momentum. This
process is depicted in Figure 1.
Whereas the wave-momentum transferred (21) for these values is (
) 313 +
577.1= of the original momentum. The wave-momentum retained by
‘negative’
photon is ( )31− times the original momentum, pointing in
opposite direction to
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A NEW MECHANICS OF CORPUSCULAR-WAVE …
235
wave-momentum of original photon. This also factually matches
that inside NRM phase velocity is opposite to the energy flow or
group velocity [4-6, 13-18, 25, 26].
The case where ,1−=pn ,1=gn (hypothetically if it exists) the
wave
momentum transferred (21) to the medium is twice the original
wave-momentum, and no mechanical momentum gets transferred to the
media, well this is case of total internal reflection. For a medium
1=pn and ,1=gn the wave and mechanical
momentum transferred to the medium is zero, that is, all the
momentum is retained by photon.
This contradiction is embedded in principle of theory of
‘wave-particle’ duality. Really if we consider photon as particle
that its linear momentum will be mvp =
and this mechanical momentum is proportional to velocity. But
while considering photon or radiation as wave then its linear
momentum is vkhp ω==λ=
here, is inversely proportional to velocity. This contradiction
is unessential if the medium is free space dispersionless vacuum
(where ) ,cv = but brings about certain problem if the photon is
inside a media (positive refractive indexed or negative refractive
indexed). We can confess therefore that value of linear momentum of
photon carrying radiation energy packet at present is far from
being established concept.
7. Derivation of Expression for Wave-momentum of Electromagnetic
Pulse ‘a Photon’
This section will elucidate the choice of our definition of wave
momentum for photon as in (17). Let a photon pulse be travelling in
free space. Observer sitting on the crest and another observer
sitting on the envelope, travelling in free space they will find
themselves at rest with respect to each other, while the packet
enters the NRM, the two observers will find that they are moving
away from each other. This is this nature of wave-momentum that is
generator of infinitesimal translations, and the infinitesimal
translations of the ‘waves’ correspond to motion of its crests and
troughs, and in NRM ‘opposes’ the direction of motion of radiation.
It is for this reason the wave-momentum points in the opposite
direction to the mechanical momentum inside NRM. Perhaps due to
this reason one may state that photon is transformed to ‘negative’
photon inside NRM, its characteristics is different than that of
original photon.
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SHANTANU DAS et al.
236
Consider photon travelling in free space with mechanical energy
,mE that is,
energy associated with its corpuscular part, and with phase or
wave-momentum as cp 0ω= having wave energy as ,pc thus total energy
is E, having relation as
(23) below, [19].
.42222 cmcpE += (23)
Call pv as phase velocity and gv as group velocity of
monochromatic EM signal
travelling in the region ( ),20 dz
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Let us associate gc as group velocity of the wave-packet, then
in above
paragraph the expression for time will be .gcZt = Let the phase
velocity
associated to crest and trough be identified as wave-velocity as
,pc then wave
momentum correlation will be ,pcEp = this makes the accompanying
mass as
.gpccEm = This validates our choice of multiplier ,2
gpvvc = and this could be
physical interpretation also.
Now for negative indexed material NRM (lossless and ideal case,
with ),1−=pn we can write an approximate relation (25), for region
(( )
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SHANTANU DAS et al.
238
( ) ( )2222 gpgp vvmvvpE +=
.22
2⎥⎦
⎤⎢⎣
⎡+=
mpvvvvm gpgp (27)
Equation (27) is for free-space, medium with positive phase and
group velocity and both equal to c. That is .cvv gp ==
Now we use (27), for NRM medium and manipulate as below:
42222 cmcpE +=
( ) ( ) ( ) ( )gpgpgpgp vvpvvmvvmvvp 222222 −=−+−=
.22
2⎥⎦
⎤⎢⎣
⎡−=
mpvvvvm gpgp (28)
Putting 2cgp ≅νν in equation (28), we get
( ) ( ) ( ).224222
2222
2222 cpcmmpccm
mpccmE −+=⎥
⎦
⎤⎢⎣
⎡−=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−= (29)
We split (29) into two parts, the mechanical (corpuscular)
energy part ( )42cm
and the energy transport by wave-momentum part ( ).22cp−
Equation (29) shows that particle energy is retained itself by
the particle, inside NRM where the phase velocity is opposite to
group velocity. In this case no (mechanical-corpuscular) energy is
transferred to the NRM medium. This we derive
from the part of rest mass-energy that is the first part of
expression ;2mc meaning
that corpuscular energy by photon is retained.
But the intriguing question is the energy due to wave-momentum
part is imaginary, inside NRM! That is equal to ipc (considering
the positive root). We can
ascribe to this imaginary ‘negative’- photon’ a wave-momentum a
value .0 cω−
Now we retard the group velocity to ,3c and have phase reversal
with phase
velocity inside NRM as ,c− then ,32cvv gp = and put the same in
(26) to get
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A NEW MECHANICS OF CORPUSCULAR-WAVE …
239
⎥⎦
⎤⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛= 2
22222
33 mpccmE
( ) ( ).31
913
91 22422242 cpcmcpcm −+=−= (27a)
Here the particle inside the NRM has less total corpuscular
energy; the difference of energy has been absorbed by the media
itself. Expression (27a)
suggests one third of the corpuscular energy ( ) 231 mc is
retained by the ‘photon’
inside the NRM slab, and the two thirds of its corpuscular
energy are given to the slab!!
Well the energy due wave momentum of the photon manifests as
imaginary
energy in this case as ( ) ,31 pci (again retaining the positive
root). We can ascribe
to this imaginary ‘negative’-photon’ a wave-momentum, a value (
) .31 0 cω−
The momentum transfer cases we have discussed in earlier section
also and maps correctly with the total energy argument cases as
described here.
Well let us consider the length of NRM slab, as ( ) ( ) ,223 Zdd
=− with pn
,1−= and .3=gn The photon is retarded in comparison to its
position in absence
of medium by distance z, which is
( ) ( ) ,1 ZnvZvcz gg
g −=−= (28a)
where Z is the thickness of medium. The relativistic form of
Newton’s first law of motion requires that the centre-of-mass
energy of a system not subjected to any external force should be
stationary or in uniform motion. Our medium is isolated from such
external influence, then the relevant total energy is sum of photon
energy
0ω and the rest mass energy of the medium ,2Mc where M is mass
of medium.
The fact that photon has been retarded by the medium means the
centre-of-mass-energy can only have been in uniform motion if the
medium has itself moved to the right by a distance ,z∆ then the
moments are
( ) ( ) ( ) ( ).02 ω=∆ zMcz (29a)
Substituting value of z from (28a), we get
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SHANTANU DAS et al.
240
( ).120 −
ω=∆ gn
McZ
z (30)
This motion can only take place if energy transfer takes place
from photon whilst inside the medium. The required velocity of the
medium is ( ) ,Zzvg ∆ from
which we can readily obtain momentum
,321 0
0medium pc
vcZ
zMvp gg =⎟⎟⎠
⎞⎜⎜⎝
⎛−
ω=∆= (31)
where the 0p is the initial momentum of the photon in free
space. Momentum
conservation suggests that we ascribe the difference between the
initial momentum and this medium momentum to the photon momentum
inside the medium. From previous section the mechanical momentum of
photon in this NRM would be
,31
31
002
02
02
1NRM p
ccnvcnnp
pggpm=
ω=ω=ω= (32)
.31
31
002
002NRM p
ccvcnp ggm =
ω=ω=ω= (33)
The wave momentum of photon inside this NRM slab is
( )
.3
1sgn0
0NRM pcnn
np
gp
pc −=
ω= (34)
Equations (32) and (33) state that ( )31 of the mechanical
momentum is
retained by the ‘photon’ inside this NRM. This is well equating
as if 1/3 of ‘particular’ photon corpuscular energy is retained by
photon inside NRM, whereas
the wave-momentum retained by photon inside NRM (34) is ( )31−
times the
original wave momentum.
8. Imaginary ‘Reactive Energy’ and ‘Wave-momentum’ inside
Medium
In the previous section, we could balance the retardation effect
stating that the corpuscular energy that comprising of mechanical
photon momentum is transferred to the medium thereby inside NRM the
retardation of photon takes place. What was intriguing was
imaginary energy of the photon inside the NRM, what we termed
as
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241
‘reactive’ energy. This reactive energy of photon inside NRM is
making the waves of phases travel backward inside NRM as contrary
to positive indexed material.
Could we reframe the wave-momentum inside a media be it positive
indexed or be it negative indexed as we have defined in (17);
rewritten as in (35)? Well the discussion suggests yes why not!
( ) ( )
.sgnsgn
00def p
nn
ncnn
np
gp
p
gp
pc =
ω (35)
A new way to define canonical momentum inside slab, be it
positive refractive indexed or negative refractive indexed system,
also this agrees with what we derived from total energy balance
description in the previous section.
Figure 1. Propagation of electromagnetic pulse. A. Pulse
propagating towards right in free space, having envelope (dashed)
and phases (solid) travelling with velocity c in same direction. B.
The same pulse touches the media with NRM with phase index as –1,
and group index as +3; shows that at the boundary there is ‘cusp’
formation and envelope retards. Here the phases travel in opposite
direction and the group (envelope) travels in same direction. This
cusp oscillates at the surface of the NRM boundary. C. The pulse is
travelling as envelope with squeezed envelope inside NRM towards
the right direction with velocity 3c+ whereas the phases are
travelling opposite to envelope, with velocity .c− The pulse is
sharpened and
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242
squeezed. This is ideal case of loss-less NRM while lossy
structures will have attenuated pulse as it travels.
9. Wave Equation Explanation
We can identify the motion of the photon pulse with mechanical
momentum but the wave momentum corresponds rather to motion of the
phase fronts. The difference is analogous to that between phase and
group velocities for a wave; the phase velocity is that at which
the phase front propagate, while the pulse and its associated
energy propagate at group velocity, thus the phase velocity does
not appear in mechanical momentum expressions used above.
We now resort to classical wave as photon and see if we can
distinguish between positive refractive indexed media and negative
refractive indexed media, through wave equation.
(a) Classical quantum prescriptor and Schrodinger wave
equation
Total energy of system is expressed as kinetic plus potential
as
.2
2EV
mpVT =+=+ (36)
By putting standard Q prescriptors that is ∇→ ip and ( ),tiE ∂∂→
and in addition asking these prescriptors to operate on wave
function ,ψ the standard
Schrodinger wave equation is obtained as
.2
22
tiV
m ∂ψ∂=ψ+ψ∇− (37)
The plane wave solution in vector form is ..1exp ⎟⎠⎞⎜
⎝⎛−=ψ rpiA
With kp = as photon’s momentum vector linked with its wave
vector, and
,ω=E without any potential the wave travels in straight line and
we have
( )0as22 == VmpE and we obtain potential free wave equation
as
.02 2
22=ψ+ψ
∂
∂ Exm
(38)
This has two solutions
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A NEW MECHANICS OF CORPUSCULAR-WAVE …
243
( )22 22 mEixmEix BeAex −+=ψ (39)
case for positive E propagating case.
( )22 22 mExmEx BeAex −+=ψ (40)
case for negative E bounded case. This bounded case is for
surface wave happens for ENG or MNG only.
(b) A new quantum prescriptor and Schrodinger wave equation
Let us take the Q prescriptors modified as
( ) ( )[ ].exp,,exp θ→ω→∂∂θ−−→ ikpEx
iip
Put them in potential free energy expression ,22 mpE = when we
operate this on
wave function ,ψ we get a new Schrodinger equation as
.02 2
222 =ψ+ψ
∂
∂θ− Exm
e i (41)
Well the solutions are for this wave equation then
( ) ( ) ( )θ−θ +=ψ imEiximEix BeAex exp2exp222
( ) ( ).expexp θ−θ += iixkiixk BeAe (42)
(42) is case for propagating case.
( ) ( ) ( )θ−θ +=ψ imEximEx BeAex exp2exp222
( ) ( ).expexp θ−θ += ixkixk BeAe (43)
(43) is case for bounded case.
A quick verification shall state that for ,0=θ one gets wave
equation for
normal media where the Right Handed Media (RHM), while π=θ gives
a wave propagation in Left Handed Media (LHM) with NRM. This also
opens up a possibility of having a system in between RHM and
LHM.
This gives a wave description of RHM and LHM where in the later
case the
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phase is opposite the energy flow can be represented as
different quantum prescriptors and different Schrodinger wave
equations. At least mathematics hints so; well physical
consequences are far from reality, at present for these new
Q-prescriptors. The rotational component ( )θiexp may be
personified as demarcation between phase velocity and group
velocity and their relation to the phase and group indices, a
future work! The future work shall also relate the relation between
this
rotational component with that of gpp nnnN sgn= in new
formulation of the
canonical (wave) momentum.
10. Conclusion
Experimental realization of negative index of refraction has as
a result raised important questions about the validity of this
negative value in well-known formulas of physics. The question of
corpuscular energy transport inside negative indexed material,
formation of reactive (imaginary) energy inside the negative
indexed substances, well the character of photon pulse especially
its momentum (corpuscular and wave) is addressed along with duality
of particle-wave nature of photon. Few new concepts regarding new
wave-momentum inside slab and reactive energy inside negative
indexed material, and new generalized wave equation is proposed to
meet the future theoretical advances on these realized negative
indexed materials.
Acknowledgement
This work is supported fully by Board of Research in Nuclear
Science (BRNS), Department of Atomic Energy (DAE).
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